# Capacity expansion

This is an augmentation to any mathematical program that has constraints representing a capacity limit. Suppose $LaTeX: g(x) \le b$ represents the limitation of using more than $LaTeX: b$ units of capacity, where $LaTeX: g(x)$ is the actual amount used for policy $LaTeX: x$. Then, this becomes a capacity expansion model by replacing the constraint with
$LaTeX: g(x) - vC \le b \;\mbox{ and }\; 0 \le v \le 1.$
Here $LaTeX: C$ is the maximum amount of new capacity, with $LaTeX: vC$ the portion brought online by choice of $LaTeX: v$. This is also reflected in the objective function with an added cost term, say $LaTeX: F(v)$, such that $LaTeX: F(0)=0$ and $LaTeX: F$ increasing on $LaTeX: [0, 1]$. If continuous use of the capacity is not possible (i.e., one builds a new plant or not), the model further requires $LaTeX: v$ to be binary-valued (0 or 1), and $LaTeX: F(1)$ is called the fixed charge.